Homogeneous algebras on the circle. I. Ideals of analytic functions

Bennett, Colin; Gilbert, John E.

Annales de l'Institut Fourier, Tome 22 (1972) no. 3, pp. 1-19.

On désigne par $𝒜$ une algèbre de Banach homogène sur le cercle et par ${𝒜}^{+}$ la sous-algèbre fermée de $𝒜$ constituée par les fonctions qui ont des prolongements analytiques dans le disque ouvert $D$. Ce travail considère la structure des idéaux fermés de ${𝒜}^{+}$, sous des restrictions convenables sur les propriétés de synthèse de $𝒜$. En particulier, on caractérise complètement les idéaux fermés de ${𝒜}^{+}$ tels que les “zero sets” rencontrent le cercle en un ensemble dénombrable. Ces résultats contiennent des résultats précédents de Kahane et de Taylor-Williams obtenus indépendamment.

Let $𝒜$ be a homogeneous algebra on the circle and ${𝒜}^{+}$ the closed subalgebra of $𝒜$ of functions having analytic extensions into the unit disk $D$. This paper considers the structure of closed ideals of ${𝒜}^{+}$ under suitable restrictions on the synthesis properties of $𝒜$. In particular, completely characterized are the closed ideals in ${𝒜}^{+}$ whose zero sets meet the circle in a countable set of points. These results contain some previous results of Kahane and Taylor-Williams obtained independently.

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